Abstract :
A local antimagic labeling of a connected graph G with at least three vertices, is a bijection f : E(G) → {1, 2, . . . , |E(G)|} such that for any two adjacent vertices u and v of G, the condition ωf (u) ̸= ωf (v) holds; where ωf (u) = ∑ x∈N(u) f(xu). Assigning ωf (u) to u for each vertex u in V (G), induces naturally a proper vertex coloring of G; and |f| denotes the number of colors appearing in this proper vertex coloring. The local antimagic chromatic number of G, denoted by χla(G), is defined as the minimum of |f|, where f ranges over all local antimagic labelings of G. In this paper, we explicitly construct an infinite class of connected graphs G such that χla(G) can be arbitrarily large while χla(G∨ K¯ 2) = 3, where G ∨ K¯ 2 is the join graph of G and the complement graph of K2. The aforementioned fact leads us to an infinite class of counterexamples to a result of [Local antimagic vertex coloring of a graph, Graphs and Combinatorics 33 (2017), 275–285].
